Extensive measurement with incomparability |
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Authors: | Erik Carlson |
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Affiliation: | Department of Philosophy, Uppsala University, Box 627, 751 26 Uppsala, Sweden |
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Abstract: | Standard theories of extensive measurement assume that the objects to be measured form a complete order with respect to the relevant property. In this paper, representation and uniqueness theorems are presented for a theory that departs radically from this completeness assumption. It is first shown that any quasi-order on a countable set can be represented by vectors of real numbers. If such an order is supplemented by a concatenation operator, yielding a relational structure that satisfies a set of axioms similar to the standard axioms for an extensive structure, we obtain a scale possessing the crucial properties of a ratio scale. Incomparability is thus compatible with extensive measurement. The paper ends with a brief discussion on some possible applications and developments of this result. |
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